The biggest counter question is identifying what constitutes high value transactions. Bitcoin works on an Unspent Transaction Outputs (UTXO) based model rather than an account based model. The input of Bitcoin transaction needs to include outpoint (
vout) of the transaction from which you control bitcoins with your keys. This input is then consumed in its entirety and any change is sent back to yourself. This is illustrated below:
/ ----- Pay for your coffee (1 BTC)
Consume UTXO (3 BTC)
\ ----- Send the 2 BTC change to yourself
Let us assume miners set aside a limit that transactions above 1.5 BTC will be charged higher fees. Considering the above cited example, should miners treat it as high value? The user paid 1 BTC for coffee (which would be below the limit) and is sending 2 BTC change back to itself (which would be above the limit). There is no sure shot way for miner to know which output was being used to spend for coffee. Although there are some methods which you can use to determine which output is a spending output versus what is a change, they are not reliable, and can be gamed by changing the amounts of outputs slightly. Moreover, how do you charge CoinJoin transaction?
If you are thinking any output value (be it spending or change) which is above a certain limit will be charged higher fees, then it will encourage users to create smaller value UTXOs so that they do not have to send higher value change back to themselves (to avoid higher fees). UTXO storage is costly as it is cached in the RAM, and to dodge higher fees users will create multiple UTXOs thus costing the network as a whole to maintain such a big costly database.
From an economic perspective I would like to point out that the most efficient way to maximize your income from mining is by using fee/vbyte. This is because Bitcoin's block size is capped, which means maximum profit can be derived from fitting in as many transactions as possible (thus dependent on size). If you try to use any other methodology, then you are creating an opportunity for other miners to step in and claim the profit your are forgoing.
This would be a classic example of economic equilibrium where any inefficiencies will be ironed out by market forces thus forcing it settle to the most practical equilibrium (fee/vbyte). Thus doing anything apart from charging fee/vbyte would require cooperation of ALL miners or a 51% attack. The former is difficult to achieve from an economic standpoint as discussed above.